Global Patterns From Local Interactions: A Dynamical Systems Approach

Abstract

In this paper the global patterns that result from local interactions between players on a two-dimensional lattice are studied. The assumptions on interaction between players are based on the Prisoner's Dilemma game that has been used extensively in game theory and in the study of biological systems. Each player is located on a square lattice, and is assumed to cooperate or defect, based on mimicking the neighbor with the highest cumulative score from the preceding round of play. The edges of the lattice are glued to form a torus. Computer simulations are conducted for different sized lattices, different payoff values, and different initial conditions. Though the paper is primarily concerned with player behavior without self-interaction, some results with self-interaction are also included. The influence of "ideal" cooperators on the evolution of the system dynamics is also studied. Three generic regimes of behavior are identified. Complex global patterns with complicated dynamics and sometimes unpredictable results occur. Steady-state solutions, simple and complex periodic solutions, and traveling waves are observed depending on the initial conditions and the payoff values.

A Dynamic Rule In Cellular Automata

“…The F-rule is an outgrowth of the Prisoner's Dilemma game as it is stated by Nowak et al [Nowak and May(1992)]. In fact, the very first introduction of the F-rule was in [Wiederien and Udwadia(2004)]. The present paper can be considered as a continuation of [Wiederien and Udwadia(2004)] where we consider more general games played among the neighbors.…”

“…In fact, the very first introduction of the F-rule was in [Wiederien and Udwadia(2004)]. The present paper can be considered as a continuation of [Wiederien and Udwadia(2004)] where we consider more general games played among the neighbors. We consider the model as a discrete dynamical system evolving in a discrete state space.…”